p−p, p−Λ, and Λ−Λ correlations studied via femtoscopy in pp reactions at √s = 7 TeV
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This is a self-archived version of an original article. This version may differ from the original in pagination and typographic details. Author(s): Title: Year: Version: Copyright: Rights: Rights url: Please cite the original version: CC BY 4.0 https://creativecommons.org/licenses/by/4.0/ p−p, p−Λ, and Λ−Λ correlations studied via femtoscopy in pp reactions at √s = 7 TeV ©2019 CERN, for the ALICE Collaboration Published version ALICE Collaboration ALICE Collaboration. (2019). p−p, p−Λ, and Λ−Λ correlations studied via femtoscopy in pp reactions at √s = 7 TeV. Physical Review C, 99(2), Article 024001. https://doi.org/10.1103/PhysRevC.99.024001 2019
PHYSICAL REVIEW C 99, 024001 (2019) p-p,p-,and-correlations studied via femtoscopy in pp reactions at √s=7TeV S. Acharya et al.∗ (ALICE Collaboration) (Received 29 June 2018; revised manuscript received 19 November 2018; published 13 February 2019) We report on the first femtoscopic measurement of baryon pairs, such as p-p,p-,and-, measured by ALICE at the Large Hadron Collider (LHC) in proton-proton collisions at √s=7 TeV. This study demonstrates the feasibility of such measurements in pp collisions at ultrarelativistic energies. The femtoscopy method is employed to constrain the hyperon-nucleon and hyperon-hyperon interactions, which are still rather poorly understood. A new method to evaluate the influence of residual correlations induced by the decays of resonances and experimental impurities is hereby presented. The p-p,p-,and-correlation functions were fitted simultaneously with the help of a new tool developed specifically for the femtoscopy analysis in small colliding systems: Correlation Analysis Tool using the Schrödinger equation (CATS). Within the assumption that in pp collisions the three particle pairs originate from a common source, its radius is found to be equal to r0=1.125 ±0.018 (stat)+0.058 −0.035 (syst) fm. The sensitivity of the measured p-correlation is tested against different scattering parameters, which are defined by the interaction among the two particles, but the statistics is not sufficient yet to discriminate among different models. The measurement of the -correlation function constrains the phase space spanned by the effective range and scattering length of the strong interaction. Discrepancies between the measured scattering parameters and the resulting correlation functions at LHC and RHIC energies are discussed in the context of various models. DOI: 10.1103/PhysRevC.99.024001 I. INTRODUCTION Traditionally femtoscopy is used in heavy-ion collisions at ultrarelativistic energies to investigate the spatial-temporal evolution of the particle emitting source created during the collision [1,2]. Assuming that the interaction for the employed particles is known, a detailed study of the geometrical extension of the emission region becomes possible [3–10]. If one considers smaller colliding systems such as protonproton (pp) at TeV energies and assumes that the particle emitting source does not show a strong time dependence, one can reverse the paradigm and exploit femtoscopy to study the final-state interaction (FSI). This is especially interesting in the case where the interaction strength is not well known as for hyperon-nucleon (Y-N) and hyperon-hyperon (Y-Y) pairs [11–19]. Hyperon-nucleon and hyperon-hyperon interactions are still rather poorly experimentally constrained and a detailed knowledge of these interactions is necessary to understand quantitatively the strangeness sector in the lowenergy regime of quantum-chromodynamics (QCD) [20]. Hyperon-nucleon (p-and p-) scattering experiments have been carried out in the 1960s [21–23] and the measured cross sections have been used to extract scattering lengths ∗Full author list given at the end of the article. Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article’s title, journal citation, and DOI. and effective ranges for the strong nuclear potential by means of effective models such as the extended-soft-core (ESC08) baryon-baryon model [24] or by means of chiral effective field theory (χEFT) approaches at leading order (LO) [25] and next-to-leading order (NLO) [26]. The results obtained from the above-mentioned models are rather different, but all confirm the attractiveness of the -nucleon (-N) interaction for low hyperon momenta. In contrast to the LO results, the NLO solution claims the presence of a negative phase shift in the p-spin singlet channel for momenta larger than p>600 MeV/c. This translates into a repulsive core for the strong interaction evident at small relative distances. The same repulsive interaction is obtained in the p-wave channel within the ESC08 model [24]. The existence of hypernuclei [27] confirms that the N- is attractive within nuclear matter for densities below nuclear saturation ρ0=0.16 fm−3. An average value of U(ρ= ρ0,k=0) ≈−30 MeV [27], with kthe hyperon momentum in the laboratory reference system, is extracted from hypernuclear data on the basis of a dispersion relation for hyperons in a baryonic medium at ρ0. The situation for the hyperon is currently rather unclear. There are some experimental indications for the formation of hypernuclei [28,29] but different theoretical approaches predict both attractive and repulsive interactions depending on the isospin state and partial wave [24,26,30]. The scarce experimental data for this hypernucleus prevents any validation of the models. A-hypernucleus candidate was detected [31] and ongoing measurements suggest that the N-interaction is weakly attractive [32]. A recent work by the Lattice HAL-QCD 2469-9985/2019/99(2)/024001(21) 024001-1 ©2019 CERN, for the ALICE Collaboration
S. ACHARYA et al. PHYSICAL REVIEW C 99, 024001 (2019) Collaboration [33] shows how this attractive interaction could be visible in the p-femtoscopy analysis, in particular by comparing correlation functions for different static source sizes. This further motivates the extension of the femtoscopic studies from heavy ions to pp collisions since in the latter case the source size decreases by about a factor of three at the LHC energies leading to an increase in the strength of the correlation signal [34]. If one considers hyperon-hyperon interactions, the most prominent example is the -case. The H-dibaryon - bound state was predicted [35] and later a double hypernucleus was observed [36]. From this single measurement a shallow -binding energy of few MeV was extracted, but the H-dibaryon state was never observed. Also recent lattice calculations [37] obtain a rather shallow attraction for the -state. The femtoscopy technique was employed by the STAR Collaboration to study -correlations in Au-Au collisions at √sNN =200 GeV [16]. First a shallow repulsive interaction was reported for the -system, but in an alternative analysis, where the residual correlations were treated more accurately [38], a shallow attractive interaction was confirmed. These analyses demonstrate the limitations of such measurements in heavy-ion collisions, where the source parameters are time dependent and the emission time might not be the same for all hadron species. The need for more experimental data to study the hyperonnucleon, hyperon-hyperon, and even the hyperon-nucleon- nucleon interaction has become more crucial in recent years due to its connection to the modeling of astrophysical objects such as neutron stars [39–42]. In the inner core of these objects the appearance of hyperons is a possible scenario since their creation at finite density becomes energetically favored in comparison with a purely neutron matter composition [41]. However, the appearance of these additional degrees of freedom leads to a softening of the nuclear matter equation of state (EOS) [43] making the EOS incompatible with the observation of neutron stars as heavy as two solar masses [44,45]. This goes under the name of the hyperon puzzle. Many attempts were made to solve this puzzle, e.g., by introducing three-body forces for NN leading to an additional repulsion that can counterbalance the large gravitational pressure and finally allow for larger neutron star masses [46–49]. A repulsive core for the two-body forces would also stiffen the EOS containing hyperons. In order to constrain the parameter space of such models a detailed knowledge of the hyperon-nucleon, including and states, and of the hyperon-nucleon-nucleon interaction is mandatory. This work presents an alternative to scattering experiments, using the femtoscopy technique to study the Y-N and Y-Y interactions in pp collisions at √s=7 TeV. We show that pp collisions at the LHC are extremely well suited to investigate baryon-baryon final-state interactions and that the measurement of the correlation function is not contaminated with the minijet background visible in meson-meson correlations [50,51]. The extracted p-p,p-, and -correlations have been compared to the predicted function obtained by solving the Schrödinger equation exactly by employing the Argonne v18 potential [52]forp-ppairs and different scattering parameters available in the literature for p-and -pairs. The predictions for the correlation function used to fit the data are obtained with the newly developed CATS framework [53]. A common source with a constant size is assumed and the value of the radius is extracted. The paper is organized in the following way. In Sec. II the experiment setup and the analysis technique are briefly introduced. In Sec. III the femtoscopy technique and the theoretical models employed are discussed. In Sec. IV the sources of systematic uncertainties are summarized and finally in Sec. V the results for the p-p,p-, and -correlation function are presented. II. DATA ANALYSIS In this paper we present results from studies of the p-p, p-, and -correlations in pp collisions at √s=7TeV employing the data collected by ALICE in 2010 during the LHC Run 1. Approximately 3.4×108minimum bias events have been used for the analysis, before event and track selection. A detailed description of the ALICE detector and its performance in the LHC Run 1 (2009-2013) is given in Refs. [54,55]. The inner tracking system (ITS) [54] consists of six cylindrical layers of high-resolution silicon detectors placed radially between 3.9 and 43 cm around the beam pipe. The two innermost layers are silicon pixel detectors (SPD) and cover the pseudorapidity range |η|<2. The time projection chamber (TPC) [56] provides full azimuthal coverage and allows charged particle reconstruction and identification (PID) via the measurement of the specific ionization energy loss dE/dx in the pseudorapidity range |η|<0.9. The time-of-flight (TOF) [57] detector consists of multigap resistive plate chambers covering the full azimuthal angle in |η|<0.9. The PID is obtained by measuring the particle’s velocity β. The above-mentioned detectors are immersed in a B=0.5 T solenoidal magnetic field directed along the beam axis. The V0 are small-angle plastic scintillator detectors used for triggering and placed on either side of the collision vertex along the beam line at +3.3 m and −0.9mfromthe nominal interaction point, covering the pseudorapidity ranges 2.8<η<5.1 (V0-A) and −3.7<η<−1.7 (V0-C). A. Event selection The minimum bias interaction trigger requires at least two out of the following three conditions: two pixel chips hit in the outer layer of the silicon pixel detectors, a signal in V0-A, a signal in V0-C [55]. Reconstructed events are required to have at least two associated tracks and the distance along the beam axis between the reconstructed primary vertex and the nominal interaction point should be smaller than 10 cm. Events with multiple reconstructed SPD vertices are considered as pileup. In addition, background events are rejected using the correlation between the number of SPD clusters and the tracklet multiplicity. The tracklets are constrained to the primary vertex, and hence a typical background event is characterized by a large amount of SPD clusters but only few tracklets, while a pileup event contains a larger number of 024001-2
p-p,p-, AND -… PHYSICAL REVIEW C 99, 024001 (2019) TABLE I. Proton (top) and candidate (bottom) selection criteria. Selection criterion Value Proton selection criteria Pseudorapidity |η|<0.8 Transverse momentum 0.5<pT<4.05 GeV/c TPC clusters nTPC >80 Crossed TPC pad rows ncrossed >70 (out of 159) Findable TPC clusters ncrossed/nfindable >0.83 Tracks with shared TPC clusters rejected Distance of closest approach xy |DCAxy|<0.1cm Distance of closest approach z|DCAz|<0.2cm Particle identification |nσ,TPC|<3forp<0.75 GeV/c nσ,combined <3forp>0.75 GeV/c Lambda selection criteria Daughter track selection criteria Pseudorapidity |η|<0.8 TPC clusters nTPC >70 Distance of closest approach DCA >0.05 cm Particle identification |nσ,TPC|<5 V0selection criteria Transverse momentum pT>0.3 GeV/c decay vertex |ivertex|<100 cm, i=x,y,z Transverse radius of the decay vertex rxy 0.2<rxy <100 cm DCA of the daughter tracks at the decay vertex DCA(|p,π|)<1.5cm Pointing angle αcos α>0.99 K0rejection 0.48 <Mπ+π−<0.515 GeV/c2 selection |Mpπ−M,PDG|<4MeV/c2 clusters at the same tracklet multiplicity. After application of these selection criteria, about 2.5×108events are available for the analysis. B. Proton candidate selection To ensure a high-purity sample of protons, strict selection criteria are imposed on the tracks. Only particle tracks reconstructed with the TPC without additional matching with hits in the ITS are considered in the analysis in order to avoid biases introduced by the nonuniform acceptance in the ITS. However, the track fitting is constrained by the independently reconstructed primary vertex. Hence, the obtained momentum resolution is comparable to that of globally reconstructed tracks, as demonstrated in Ref. [55]. The selection criteria for the proton candidates are summarized in Table I. The selection on the number of reconstructed TPC clusters serves to ensure the quality of the track, to assure a good pTresolution at large momenta and to remove fake tracks from the sample. To enhance the number of protons produced at the primary vertex, a selection is imposed on the distance of closest approach (DCA) in both beam (z) and transverse (xy) directions. In order to minimize the fraction of protons originating from the interaction of primary particles with the detector material, a low transverse momentum cutoff is applied [58]. At high pTa cutoff is introduced to ensure the purity of the proton sample, as the purity drops below 80% for larger pTdue to the decreasing separation power of the combined TPC and TOF particle identification. For particle identification both the TPC and the TOF detectors are employed. For low momenta (p<0.75 GeV/c) only the PID selection from the TPC is applied, while for larger momenta the information of both detectors is combined since the TPC does not provide a sufficient separation power in this momentum region. The combination of TPC and TOF signals is done by employing a circular selection criterion nσ,combined ≡(nσ,TPC)2+(nσ,TOF )2, where nσis the number of standard deviations of the measured from the expected signal at a given momentum. The expected signal is computed in the case of the TPC from a parametrized Bethe-Bloch curve, and in the case of the TOF by the expected βof a particle with a mass hypothesis m. In order to further enhance the purity of the proton sample, the nσis computed assuming different particle hypotheses (kaons, electrons, and pions) and if the corresponding hypothesis is found to be more favorable, i.e., the nσvalue found to be smaller, the proton hypothesis and thus the track is rejected. With these selection criteria apT-averaged proton purity of 99% is achieved. The purity remains above 99% for pT<2GeV/cand then decreases to 80% at the momentum cutoff of 4.05 GeV/c. C. candidate selection The weak decay →pπ−(BR =63.9%, cτ=7.3cm [59]) is exploited for the reconstruction of the candidate, and accordingly the charge-conjugate decay for the identification. The reconstruction method forms so-called V0 decay candidates from two charged particle tracks using a procedure described in Ref. [60]. The selection criteria for 024001-3
S. ACHARYA et al. PHYSICAL REVIEW C 99, 024001 (2019) the candidates are summarized in Table I.TheV0daughter tracks are globally reconstructed tracks and, in order to maximize the efficiency, selected by a broad particle identification cut employing the TPC information only. Additionally, the daughter tracks are selected by requiring a minimum impact parameter of the tracks with respect to the primary vertex. After the selection all positively charged daughter tracks are combined with a negatively charged partner to form a pair. The resulting vertex ivertex,i=x,y,zis then defined as the point of closest approach between the two daughter tracks. This distance of closest approach of the two daughter tracks with respect to the decay vertex DCA(|p,π|)isusedasan additional quality criterion of the candidate. The momentum is calculated as the sum of the daughter momenta. A minimum transverse momentum requirement on the candidate is applied to reduce the contribution of fake candidates. Finally, a selection is applied on the opening angle αbetween the momentum and the vector pointing from the primary vertex to the secondary V0decay vertex. The rather broad PID selection of the daughter tracks introduces a residual pion contamination of the proton daughter sample that in combination with the charge-conjugate pion of the V0 leads to the misidentification of K0 Sas candidates. These K0 S candidates are removed by a selection on the π+π−invariant mass. The reconstructed invariant mass, its resolution and purity are determined by fitting eight spectra of the same size in pT∈[0.3,4.3] GeV/cwith the sum of two Gaussian functions describing the signal and a second-order polynomial to emulate the combinatorial background. The obtained values for the mean and variance of the two Gaussian functions are combined with an arithmetic average. The determined mass is in agreement with the PDG value for the and particles [59]. A total statistics of 5.9×106and 5.5×106and a signal to background ratio of 20 and 25 at a pT-averaged purity of 96% and 97% is obtained for and , respectively. It should be noted that the purity is constant within the investigated pTrange. Finally, a selection on the pπ−(pπ+) invariant mass FIG. 1. Invariant mass distribution of pπ−(pπ+) to obtain the () signal. The dashed lines set the selection width used in the analysis. is applied. To avoid any contribution from autocorrelations, all candidates are checked for shared daughter tracks. If this condition is found to be true, the candidate with the smaller cosine pointing angle is removed from the sample. If a primary proton is also used as a daughter track of a candidate, the latter is rejected. Figure 1shows the pT- integrated invariant mass of the and candidates. III. CORRELATION FUNCTION The observable of interest in femtoscopy is the two-particle correlation function, which is defined as the probability to find simultaneously two particles with momenta p1and p2 divided by the product of the corresponding single-particle probabilities C(p1,p2)≡P(p1,p2) P(p1)·P(p2).(1) These probabilities are directly related to the inclusive Lorentz invariant spectra P(p1,p2)=E1E2d6N d3p1d3p2and P(p1,2)= E1,2d3N d3p1,2. In absence of a correlation signal the value of C(p1,p2) equals unity. Approximating the emission process and the momenta of the particles, the size of the particle emitting source can be studied. Following [2], Eq. (1) can then be rewritten as C(k∗)=d3r∗S(r∗)|ψ(r∗,k∗)|2,(2) where k∗is the relative momentum of the pair defined as k∗= 1 2·|p∗ 1−p∗ 2|, with p∗ 1and p∗ 2the momenta of the two particles in the pair rest frame (PRF, denoted by the ∗), S(r∗) contains the distribution of the relative distance of particle pairs in the pair rest frame, the so-called source function, and ψ(r∗,k∗) denotes the relative wave function of the particle pair. The latter contains the particle interaction term and determines the shape of the correlation function. In this work, the p-p correlation function, which is theoretically well understood, is employed to obtain the required information about the source function and this information will be used to study the p- and -interaction. In order to relate the correlation function to experimentally accessible quantities, Eq. (1) is reformulated [2]as C(k∗)=NA(k∗) B(k∗).(3) The distribution of particle pairs from the same event is denoted with A(k∗) and B(k∗) is a reference sample of uncorrelated pairs. The latter is obtained using event mixing techniques, in which the particle pairs of interest are combined from single particles from different events. To avoid acceptance effects of the detector system, the mixing procedure is conducted only between particle pairs stemming from events with similar zposition of the primary vertex and similar multiplicity [2]. The normalization parameter for mixed and same event yields Nis chosen such that the mean value of the correlation function equals unity for k∗∈[0.2,0.4] GeV/c. As correlation functions of all studied baryon-baryon pairs, i.e., p-p,p-, and -, exhibit identical behavior compared 024001-4
p-p,p-, AND -… PHYSICAL REVIEW C 99, 024001 (2019) TABLE II. Weight parameters of the individual components of the p-p,p-,and-correlation function. p-pp-- Pair λparameter [%] Pair λparameter [%] Pair λparameter [%] pp 74.18 p47.13 29.94 pp15.52 p−9.92 019.96 pp0.81 p09.92 003.33 p+p6.65 p015.71 012.61 p+p+0.15 p4.93 001.33 pp+0.70 p−1.04 −12.61 ˜pp 1.72 p01.04 −−1.33 ˜pp0.18 p01.64 004.20 ˜pp+0.08 p+2.11 0−4.20 ˜p˜p0.01 p+−0.44 0−2.65 p+00.44 ˜ 4.38 p+00.70 ˜ 01.46 ˜p0.55 ˜ 00.92 ˜p−0.18 ˜ −0.92 ˜p00.12 ˜ ˜ 0.16 ˜p00.12 p˜ 3.45 p˜ 0.36 p+˜ 0.15 ˜p˜ 0.04 to those of their respective antibaryon-antibaryon pairs, the corresponding samples are combined to enhance the statistical significance. Therefore, in the following p-pdenotes the combination of p-p⊕p-p, and accordingly for p-and -. A. Decomposition of the correlation function The experimental determination of the correlation function is distorted by two distinct mechanisms. The sample of genuine particle pairs include misidentified particles and feed-down particles from strong and weak decays. In this work a new method to separate all the individual components contributing to a measured correlation signal is proposed. The correlation functions arising from resonances or impurities of the sample are weighted with the so-called λ parameters and in this way are taken into account in the total correlation function of interest C(k∗)=1+λgenuine ·[Cgenuine(k∗)−1] + ij λij[Cij(k∗)−1],(4) where the i,jdenote all possible impurity and feed-down contributions. These λparameters can be obtained employing exclusively single-particle properties such as the purity and feed-down probability. The underlying mathematical formalism is outlined in the Appendix. For the case of p-pcorrelation the following contributions must be taken into account: {pp}=pp +pp+pp+p+p+p+p+ +pp++˜pp+˜pp+˜pp++˜p˜p,(5) where ˜ Xrefers to misidentified particles of specie X.p, and p+correspond to protons stemming from the weak decay of the corresponding hyperons. The →π →pππ decays are explicitly considered in the feed-down contribution of the p-correlation and hence are omitted in Eq. (5)to avoid double counting. As shown in Appendix, the fraction of primary protons and their feed-down fractions are required to calculate the λparameters of the different contributions to Eq. (5). The information about the origin of the protons, i.e., whether the particles are of primary origin, originating from feed down or from the interactions with the detector material, is obtained by fitting Monte Carlo (MC) templates to the experimental distributions of the distance of closest approach of the track to the primary vertex. The MC templates and the purity are extracted from PYTHIA [61] simulations using the Perugia 2011 tune [62], which were filtered through the ALICE detector and the reconstruction algorithm [54]. The pTaverages are then calculated by weighting the quantities of interest by the respective particle yields dN/dpT.The resulting fraction of primary protons averaged over pTis 87% (where in this fraction we also include the protons stemming from strong decays of broad resonances), with the other 13% of the total yield associated to weak decays of resonances and the contribution from the detector material is found to be negligible. The feed down from weakly decaying resonances is evaluated by using cross sections from PYTHIA and for the proton sample consists of the (70%) and +(30%) contributions. The individual contributions to the total correlation function are presented in Table II. The decomposition of the p-correlation function is conducted in a similar manner as for the p-ppair, however, 024001-5
S. ACHARYA et al. PHYSICAL REVIEW C 99, 024001 (2019) considering the purities and feed-down fractions of both particles {p}=p+p−+p0+p0+p+p− +p0+p0+p++p+−+p+0 +p+0+˜p+˜p−+˜p0+˜p0 +p˜ +p˜ +p+˜ +˜p˜ . (6) The purity is obtained from fits to the invariant mass spectra in eight bins of pTand defined as S/(S+B), where Sdenotes the actual signal and Bthe background. The feeddown contribution is determined from MC template fits of the experimental distributions of the cosine pointing angle, in which a total of four templates are considered corresponding to direct, feed-down, material, and impurity contributions. The production probability dN/dpTis employed in order to obtain pTweighted average values. Around 73% of the s are primaries (where in this fraction we also include the stemming from strong decays of broad resonances) and 23% originate from weakly decaying resonances, which is in line with the values quoted in Ref. [63]. The remaining yield is associated to combinatorial background and s produced in the detector material. The main contribution to the feed-down fraction is expected to originate from the states with no preference for the neutral or the charged, respectively. This hypothesis is supported by PYTHIA simulations where the secondary particles arise from the weak decay of the 0(48%) and ±(49%) resonances. The remaining contribution in the simulation arises from the 0, which, however, is treated separately. Since the latter decays electromagnetically almost exclusively into γ [59], it has a very short life time and cannot be experimentally differentiated from the sample of primary s. Measurements of the ratio R0/ =σ0/σhave obtained values around 1/3[64– 67], however, with large uncertainties for hadronic collisions at high energies. For lack of better estimates the value of 1/3 is used in the following. The resulting λparameters for the p-pair are shown in Table II. For the -correlation function the following pair contributions are taken into account: {}= +0+00+0+00 +−+−−+00+0− +0−+˜ +˜ 0+˜ − +˜ 0+˜ ˜ . (7) The resulting λparameters are shown in Table II. Notable is that the actual pair of interest contributes only to about onethird of the signal, while pair fractions involving in particular 0and give a significant contribution. The statistical uncertainties of these parameters are negligible and their influence on the systematic uncertainties will be evaluated in Sec. IV. Possible effects that were considered in this analysis and that could be influencing the source are either the decay of strong resonances, or a mTscaling. The latter is related to collective effects and may result in a non-Gaussian behavior of the source. The p-pcorrelation function is here considered as a benchmark since the theoretical description of the interaction is well established. The good agreement between the mT- integrated data and the femtoscopic fit demonstrates that the assumption of a Gaussian source is pertinent. A comparison of the mTdistribution for p-pand p-pairs shows a good agreement between the two. The limited experimental sample, however, does not allow conducting a differential analysis yet. Additionally, we have considered the effect of the strong decays such as →N+πand N∗→+Kon the production of protons and . Since the and N∗resonances have typical widths above 120 MeV the decay length is in the order of 1 fm so that the decay particles do still experience the final-state interaction with the neighboring particles as the primaries do. We estimate that about 65% of all primary protons and stem from the strong decay of resonances [68] and we have simulated how the source could be modified by such effects. A difference of 5% in the results of the Gaussian fit is found when comparing p-pto p-pairs. These effects are in the same order of the present uncertainty on the radius and are herewith neglected. A quantitative study including all resonances is planned in the analysis of the LHC Run 2 sample. B. Detector effects The shape of the experimentally determined correlation function is affected by the finite momentum resolution. This is taken into account when the experimental data are compared to model calculations in the fitting procedure by transforming the modeled correlation function, see Eq. (15), to the reconstructed momentum basis. When tracks of particle pairs involved in the correlation function are almost collinear, i.e., have a low k∗, detector effects can affect the measurement. No hint for track merging or splitting is found and therefore no explicit selection criteria are introduced. C. Nonfemtoscopic background For sufficiently large relative momenta (k∗>200 MeV/c) and increasing separation distance, the FSI among the particles is suppressed and hence the correlation function should approach unity. As shown in Fig. 2, however, the measured correlation function for p-pand p-exhibits an increase for k∗larger than about 200 MeV/cfor the two systems. Such nonfemtoscopic effects, probably due to energy-momentum conservation, are in general more pronounced in small colliding systems where the average particle multiplicity is low [2]. In the case of meson-meson correlations at ultrarelativistic energies, the appearance of long-range structures in the correlation functions for moderately small k∗(k∗<200 MeV/c)is typically interpreted as originating from minijetlike structures [50,69]. PYTHIA also shows the same nonfemtoscopic correlation for larger k∗but fails to reproduce quantitatively the behavior shown in Fig. 2, as already observed for the angular correlation of baryon-baryon and antibaryon-antibaryon pairs [58]. Energy-momentum conservation leads to a contribution to the signal, which can be reproduced with a formalism described in Ref. [70] and is accordingly also considered in 024001-6
p-p,p-, AND -… PHYSICAL REVIEW C 99, 024001 (2019) FIG. 2. The raw correlation function compared to PYTHIA6 Perugia 2011 simulations for (a) p-p,(b)p-,and(c)-pairs. this work. Therefore, a linear function C(k∗)nonfemto =ak∗+ bwhere a,bare fit parameters, is included to the global fit as C(k∗)=C(k∗)femto ×C(k∗)nonfemto to improve the description of the signal by the femtoscopic model. The fit parameters of the baseline function are obtained in k∗∈[0.3,0.5] GeV/c for p-pand p-pairs. For the case of the -correlation function, the uncertainties of the data do not allow to additionally add a baseline, which is therefore omitted in the femtoscopic fit. D. Modeling the correlation function 1. Genuine correlation function For the p-pcorrelation function the Coulomb and the strong interaction as well as the antisymmetrization of the wave functions are considered [71]. The strong interaction part of the potential is modeled employing the Argonne v18 [52] potential considering the sand pwaves. The source is assumed to be isotropic with a Gaussian profile of radius r0. The resulting Schrödinger equation is then solved with the CATS [53]. In the case of p-and -we employ the Lednický and Lyuboshitz analytical model [72] to describe these correlation functions. This model is based on the assumption of an isotropic source with Gaussian profile S(r0)=1 4πr2 03/2exp−r2 4r2 0,(8) where r0is the size of the source. Additionally, the complex scattering amplitude is evaluated by means of the effective range approximation f(k∗)S=1 fS 0+1 2dS 0k∗2−ik∗−1 ,(9) with the scattering length fS 0, the effective range dS 0and S denoting the total spin of the particle pair. In the following, the usual sign convention of femtoscopy is employed where an attractive interaction leads to a positive scattering length. With these assumptions the analytical description of the correlation function for uncharged particles [72] reads C(k∗)Lednicky =1+ S ρS1 2 f(k∗)S r0 21−dS 0 2√πr0 +2Ref(k∗)S √πr0 F1(Qinvr0)−Imf(k∗)S r0 F2(Qinvr0),(10) where Ref(k∗)S[Imf(k∗)S] denotes the real (imaginary) part of the complex scattering amplitude, respectively. The F1(Qinvr0) and F2(Qinvr0) are analytical functions resulting from the approximation of isotropic emission with a Gaussian source and the factor ρScontains the pair fraction emitted into a certain spin state S.Forthep-pair unpolarized emission is assumed. The -pair is composed of identical particles and hence additionally quantum statistics needs to be considered, which leads to the introduction of an additional term to the Lednický model, as employed, e.g., in Ref. [16]. While the CATS framework can provide an exact solution for any source and local interaction potential, the Lednicky- Lyuboshitz approach uses the known analytical solution outside the range of the strong interaction potential and takes into account its modification in the inner region in an approximate way only. That is why this approach may not be valid for small systems. 2. Residual correlations Table II demonstrates that a significant admixture of residuals is present in the experimental sample of particle pairs. A first theoretical investigation of these so-called residual correlations was conducted in Ref. [73]. This analysis relies on the procedure established in Ref. [19], where the initial correlation function of the residual is calculated and then transformed to the new momentum basis after the decay. For the p-pchannel only the feed down from the p- correlation function is considered, which is obtained by fitting the p-experimental correlation function and then transforming it to the p-pmomentum basis. All contributions are weighted by the corresponding λparameters and the modeled correlation function for this pair Cmodel,p-p(k∗) can be written 024001-7
S. ACHARYA et al. PHYSICAL REVIEW C 99, 024001 (2019) TABLE III. Selection parameter variation and the resulting relative systematic uncertainty on the p-p,p-,and-correlation function. Variable Default Variation p-p[%] p-[%] -[%] Min. pTproton (GeV/c) 0.5 0.4, 0.6 1 0.2 – |η|proton 0.8 0.7, 0.9 0.4 0.2 – nσproton 3 2, 5 1.8 0.2 – Proton tracks TPC only Global 2.4 0 – nCluster proton 80 90 0.3 0.1 – Min. pTV0(GeV/c) 0.3 0.24, 0.36 – 0 0 cos(α)V00.99 0.998 – 0 1.8 nσV0daughter 5 4 – 0.1 0.3 nCluster V0daughter 70 80 – 0.1 0.7 |η|V00.8 0.7, 0.9 – 0.6 0.8 DCA(|p,π|) (cm) 1.5 1.2 – 0.5 0 DCA (cm) 0.05 0.06 – 0.7 0.6 as Cmodel,p-p(k∗)=1+λpp ·[Cpp(k∗)−1]+λppCpp(k∗)−1. (11) All other residual correlations are assumed to be flat. For the p-, residual correlations from the p-0,p-, and -pairs are taken into account. As the -correlation function is rather flat no further transformation is applied. The p-0correlation function is obtained using predictions from Ref. [74]. As the decay products of the reaction →π are charged and therefore accessible by ALICE, we measure the p- correlation function. The experimental data are parametrized with a phenomenological function Cp−−(k∗)=1+exp(−k∗a) k∗a ,(12) where the parameter ais employed to scale the function to the data and has no physical meaning. Its value is found to be a=3.88 fm. The modeled correlation functionCmodel,p-(k∗) for the pair is obtained by Cmodel,p-(k∗)=1+λp[Cp(k∗)−1] +λp0Cp0(k∗)−1 +λp−Cp−(k∗)−1.(13) As the present knowledge on the hyperon-hyperon interaction is scarce, in particular regarding the interaction of the with other hyperons, all residual correlations feeding into the -correlation function are considered to be consistent with unity, Cmodel,-(k∗)=1+λ[C(k∗)−1].(14) It should be noted, that the residual correlation functions, after weighting with the corresponding λparameter, transformation to the momentum base of the correlation of interest and taking into account the finite momentum resolution, only barely contribute to the total fit function. 3. Total correlation function model The correlation function modeled according to the considerations discussed above is then multiplied by a linear function to correct for the baseline as discussed in Sec. III C and weighted with a normalization parameter N Ctot(k∗)=N·(a+b·k∗)·Cmodel(k∗),(15) where Cmodel(k∗) incorporates all considered theoretical correlation functions, weighted with the corresponding λparameters as discussed in Secs. III A and III D. The inclusion of a baseline is further motivated by the presence of a linear but nonflat correlation observed in the data outside the femtoscopic region (see Fig. 2 for k∗∈[0.3,0.5] GeV/c). When attempting to use a higher-order polynomial to model the background, the resulting curves are still compatible with a linear function, while their interpolation into the lower k∗region leads to an overall poorer fit quality. IV. SYSTEMATIC UNCERTAINTIES A. Correlation function The systematic uncertainties of the correlation functions are extracted by varying the proton and candidate selection criteria according to Table III. Due to the low number of particle pairs, in particular at low k∗, the resulting variations of the correlation functions are in general much smaller than the statistical uncertainties. In order to still estimate the systematic uncertainties the data are rebinned by a factor of 10. The systematic uncertainty on the correlation function is obtained by computing the ratio of the default correlation function to the one obtained by the respective cut variation. Whenever this results in two systematic uncertainties, i.e., by a variation up and downwards, the average is taken into account. Then all systematic uncertainties from the cut variations are summed up quadratically. This is then extrapolated to the finer binning of the correlation function by fitting a polynomial of second order. The obtained systematic uncertainties are found to be largest in the lowest k∗bin. The individual contributions in 024001-8
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S. ACHARYA et al. PHYSICAL REVIEW C 99, 024001 (2019) M. Shimomura,82 S. Shirinkin,64 Q. Shou,6,110 K. Shtejer,26 Y. Sibiriak,87 S. Siddhanta,54 K. M. Sielewicz,34 T. Siemiarczuk,84 D. Silvermyr,80 G. Simatovic,89 G. Simonetti,34,103 R. Singaraju,139 R. Singh,85 R. Singh,99 V. Singhal,139 T. Sinha,107 B. Sitar,14 M. Sitta,32 T. B. Skaali,21 M. Slupecki,126 N. Smirnov,144 R. J. M. Snellings,63 T. W. Snellman,126 J. Sochan,115 C. Soncco,109 J. Song,18 F. Soramel,29 S. Sorensen,128 F. Sozzi,104 I. Sputowska,117 J. Stachel,102 I. Stan,68 P. Stankus,94 E. Stenlund,80 D. Stocco,113 M. M. Storetvedt,36 P. Strmen,14 A. A. P. Suaide,120 T. Sugitate,45 C. Suire,61 M. Suleymanov,15 M. Suljic,34,25 R. Sultanov,64 M. Šumbera,93 S. Sumowidagdo,50 K. Suzuki,112 S. Swain,66 A. Szabo,14 I. Szarka,14 U. Tabassam,15 J. Takahashi,121 G. J. Tambave,22 N. Tanaka,131 M. Tarhini,113 M. Tariq,17 M. G. Tarzila,47 A. Tauro,34 G. Tejeda Muñoz,44 A. Telesca,34 C. Terrevoli,29 B. Teyssier,133 D. Thakur,49 S. 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Alikhanyan National Science Laboratory (Yerevan Physics Institute) Foundation, Yerevan, Armenia 2Bogolyubov Institute for Theoretical Physics, National Academy of Sciences of Ukraine, Kiev, Ukraine 3Bose Institute, Department of Physics and Centre for Astroparticle Physics and Space Science (CAPSS), Kolkata, India 4Budker Institute for Nuclear Physics, Novosibirsk, Russia 5California Polytechnic State University, San Luis Obispo, California, United States 6Central China Normal University, Wuhan, China 7Centre de Calcul de l’IN2P3, Villeurbanne, Lyon, France 8Centro de Aplicaciones Tecnológicas y Desarrollo Nuclear (CEADEN), Havana, Cuba 9Centro de Investigación y de Estudios Avanzados (CINVESTAV), Mexico City and Mérida, Mexico 10Centro Fermi - Museo Storico della Fisica e Centro Studi e Ricerche “Enrico Fermi”, Rome, Italy 11Chicago State University, Chicago, Illinois, United States 12China Institute of Atomic Energy, Beijing, China 13Chonbuk National University, Jeonju, Republic of Korea 14Comenius University Bratislava, Faculty of Mathematics, Physics and Informatics, Bratislava, Slovakia 15COMSATS Institute of Information Technology (CIIT), Islamabad, Pakistan 16Creighton University, Omaha, Nebraska, United States 17Department of Physics, Aligarh Muslim University, Aligarh, India 18Department of Physics, Pusan National University, Pusan, Republic of Korea 19Department of Physics, Sejong University, Seoul, Republic of Korea 20Department of Physics, University of California, Berkeley, California, United States 21Department of Physics, University of Oslo, Oslo, Norway 22Department of Physics and Technology, University of Bergen, Bergen, Norway 23Dipartimento di Fisica dell’Università ’La Sapienza’ and Sezione INFN, Rome, Italy 24Dipartimento di Fisica dell’Università and Sezione INFN, Cagliari, Italy 25Dipartimento di Fisica dell’Università and Sezione INFN, Trieste, Italy 26Dipartimento di Fisica dell’Università and Sezione INFN, Turin, Italy 27Dipartimento di Fisica e Astronomia dell’Università and Sezione INFN, Bologna, Italy 28Dipartimento di Fisica e Astronomia dell’Università and Sezione INFN, Catania, Italy 29Dipartimento di Fisica e Astronomia dell’Università and Sezione INFN, Padova, Italy 30Dipartimento di Fisica ‘E. R. Caianiello’ dell’Università and Gruppo Collegato INFN, Salerno, Italy 31Dipartimento DISAT del Politecnico and Sezione INFN, Turin, Italy 024001-18
p-p,p-, AND -… PHYSICAL REVIEW C 99, 024001 (2019) 32Dipartimento di Scienze e Innovazione Tecnologica dell’Università del Piemonte Orientale and INFN Sezione di Torino, Alessandria, Italy 33Dipartimento Interateneo di Fisica ‘M. Merlin’ and Sezione INFN, Bari, Italy 34European Organization for Nuclear Research (CERN), Geneva, Switzerland 35Faculty of Electrical Engineering, Mechanical Engineering and Naval Architecture, University of Split, Split, Croatia 36Faculty of Engineering and Science, Western Norway University of Applied Sciences, Bergen, Norway 37Faculty of Nuclear Sciences and Physical Engineering, Czech Technical University in Prague, Prague, Czech Republic 38Faculty of Science, P. J. Šafárik University, Košice, Slovakia 39Frankfurt Institute for Advanced Studies, Johann Wolfgang Goethe-Universität Frankfurt, Frankfurt, Germany 40Gangneung-Wonju National University, Gangneung, Republic of Korea 41Gauhati University, Department of Physics, Guwahati, India 42Helmholtz-Institut für Strahlen- und Kernphysik, Rheinische Friedrich-Wilhelms-Universität Bonn, Bonn, Germany 43Helsinki Institute of Physics (HIP), Helsinki, Finland 44High Energy Physics Group, Universidad Autónoma de Puebla, Puebla, Mexico 45Hiroshima University, Hiroshima, Japan 46Hochschule Worms, Zentrum für Technologietransfer und Telekommunikation (ZTT), Worms, Germany 47Horia Hulubei National Institute of Physics and Nuclear Engineering, Bucharest, Romania 48Indian Institute of Technology Bombay (IIT), Mumbai, India 49Indian Institute of Technology Indore, Indore, India 50Indonesian Institute of Sciences, Jakarta, Indonesia 51INFN, Laboratori Nazionali di Frascati, Frascati, Italy 52INFN, Sezione di Bari, Bari, Italy 53INFN, Sezione di Bologna, Bologna, Italy 54INFN, Sezione di Cagliari, Cagliari, Italy 55INFN, Sezione di Catania, Catania, Italy 56INFN, Sezione di Padova, Padova, Italy 57INFN, Sezione di Roma, Rome, Italy 58INFN, Sezione di Torino, Turin, Italy 59INFN, Sezione di Trieste, Trieste, Italy 60Inha University, Incheon, Republic of Korea 61Institut de Physique Nucléaire d’Orsay (IPNO), Institut National de Physique Nucléaire et de Physique des Particules (IN2P3/CNRS), Université de Paris-Sud, Université Paris-Saclay, Orsay, France 62Institute for Nuclear Research, Academy of Sciences, Moscow, Russia 63Institute for Subatomic Physics, Utrecht University/Nikhef, Utrecht, Netherlands 64Institute for Theoretical and Experimental Physics, Moscow, Russia 65Institute of Experimental Physics, Slovak Academy of Sciences, Košice, Slovakia 66Institute of Physics, Homi Bhabha National Institute, Bhubaneswar, India 67Institute of Physics of the Czech Academy of Sciences, Prague, Czech Republic 68Institute of Space Science (ISS), Bucharest, Romania 69Institut für Kernphysik, Johann Wolfgang Goethe-Universität Frankfurt, Frankfurt, Germany 70Instituto de Ciencias Nucleares, Universidad Nacional Autónoma de México, Mexico City, Mexico 71Instituto de Física, Universidade Federal do Rio Grande do Sul (UFRGS), Porto Alegre, Brazil 72Instituto de Física, Universidad Nacional Autónoma de México, Mexico City, Mexico 73iThemba LABS, National Research Foundation, Somerset West, South Africa 74Johann-Wolfgang-Goethe Universität Frankfurt Institut für Informatik, Fachbereich Informatik und Mathematik, Frankfurt, Germany 75Joint Institute for Nuclear Research (JINR), Dubna, Russia 76Korea Institute of Science and Technology Information, Daejeon, Republic of Korea 77KTO Karatay University, Konya, Turkey 78Laboratoire de Physique Subatomique et de Cosmologie, Université Grenoble-Alpes, CNRS-IN2P3, Grenoble, France 79Lawrence Berkeley National Laboratory, Berkeley, California, United States 80Lund University Department of Physics, Division of Particle Physics, Lund, Sweden 81Nagasaki Institute of Applied Science, Nagasaki, Japan 82Nara Women’s University (NWU), Nara, Japan 83National and Kapodistrian University of Athens, School of Science, Department of Physics, Athens, Greece 84National Centre for Nuclear Research, Warsaw, Poland 85National Institute of Science Education and Research, Homi Bhabha National Institute, Jatni, India 86National Nuclear Research Center, Baku, Azerbaijan 024001-19
S. ACHARYA et al. PHYSICAL REVIEW C 99, 024001 (2019) 87National Research Centre Kurchatov Institute, Moscow, Russia 88Niels Bohr Institute, University of Copenhagen, Copenhagen, Denmark 89Nikhef, National institute for subatomic physics, Amsterdam, Netherlands 90NRC Kurchatov Institute IHEP, Protvino, Russia 91NRNU Moscow Engineering Physics Institute, Moscow, Russia 92Nuclear Physics Group, STFC Daresbury Laboratory, Daresbury, United Kingdom 93Nuclear Physics Institute of the Czech Academy of Sciences, ˇ Rež u Prahy, Czech Republic 94Oak Ridge National Laboratory, Oak Ridge, Tennessee, United States 95Ohio State University, Columbus, Ohio, United States 96Petersburg Nuclear Physics Institute, Gatchina, Russia 97Physics department, Faculty of science, University of Zagreb, Zagreb, Croatia 98Physics Department, Panjab University, Chandigarh, India 99Physics Department, University of Jammu, Jammu, India 100Physics Department, University of Rajasthan, Jaipur, India 101Physikalisches Institut, Eberhard-Karls-Universität Tübingen, Tübingen, Germany 102Physikalisches Institut, Ruprecht-Karls-Universität Heidelberg, Heidelberg, Germany 103Physik Department, Technische Universität München, Munich, Germany 104Research Division and ExtreMe Matter Institute EMMI, GSI Helmholtzzentrum für Schwerionenforschung GmbH, Darmstadt, Germany 105Rudjer Boškovi´c Institute, Zagreb, Croatia 106Russian Federal Nuclear Center (VNIIEF), Sarov, Russia 107Saha Institute of Nuclear Physics, Homi Bhabha National Institute, Kolkata, India 108School of Physics and Astronomy, University of Birmingham, Birmingham, United Kingdom 109Sección Física, Departamento de Ciencias, Pontificia Universidad Católica del Perú, Lima, Peru 110Shanghai Institute of Applied Physics, Shanghai, China 111St. Petersburg State University, St. Petersburg, Russia 112Stefan Meyer Institut für Subatomare Physik (SMI), Vienna, Austria 113SUBATECH, IMT Atlantique, Université de Nantes, CNRS-IN2P3, Nantes, France 114Suranaree University of Technology, Nakhon Ratchasima, Thailand 115Technical University of Košice, Košice, Slovakia 116Technische Universität München, Excellence Cluster ‘Universe’, Munich, Germany 117The Henryk Niewodniczanski Institute of Nuclear Physics, Polish Academy of Sciences, Cracow, Poland 118The University of Texas at Austin, Austin, Texas, United States 119Universidad Autónoma de Sinaloa, Culiacán, Mexico 120Universidade de São Paulo (USP), São Paulo, Brazil 121Universidade Estadual de Campinas (UNICAMP), Campinas, Brazil 122Universidade Federal do ABC, Santo Andre, Brazil 123University College of Southeast Norway, Tonsberg, Norway 124University of Cape Town, Cape Town, South Africa 125University of Houston, Houston, Texas, United States 126University of Jyväskylä, Jyväskylä, Finland 127University of Liverpool, Liverpool, United Kingdom 128University of Tennessee, Knoxville, Tennessee, United States 129University of the Witwatersrand, Johannesburg, South Africa 130University of Tokyo, Tokyo, Japan 131University of Tsukuba, Tsukuba, Japan 132Université Clermont Auvergne, CNRS/IN2P3, LPC, Clermont-Ferrand, France 133Université de Lyon, Université Lyon 1, CNRS/IN2P3, IPN-Lyon, Villeurbanne, Lyon, France 134Université de Strasbourg, CNRS, IPHC UMR 7178, F-67000 Strasbourg, France, Strasbourg, France 135Université Paris-Saclay Centre d’etudes de Saclay (CEA), IRFU, Department de Physique Nucléaire (DPhN), Saclay, France 136Università degli Studi di Foggia, Foggia, Italy 137Università degli Studi di Pavia, Pavia, Italy 138Università di Brescia, Brescia, Italy 139Variable Energy Cyclotron Centre, Homi Bhabha National Institute, Kolkata, India 140Warsaw University of Technology, Warsaw, Poland 141Wayne State University, Detroit, Michigan, United States 142Westfälische Wilhelms-Universität Münster, Institut für Kernphysik, Münster, Germany 024001-20
p-p,p-, AND -… PHYSICAL REVIEW C 99, 024001 (2019) 143Wigner Research Centre for Physics, Hungarian Academy of Sciences, Budapest, Hungary 144Yale University, New Haven, Connecticut, United States 145Yonsei University, Seoul, Republic of Korea aDeceased. bDipartimento DET del Politecnico di Torino, Turin, Italy. cM. V. Lomonosov Moscow State University, D. V. Skobeltsyn Institute of Nuclear, Physics, Moscow, Russia. dDepartment of Applied Physics, Aligarh Muslim University, Aligarh, India. eInstitute of Theoretical Physics, University of Wroclaw, Poland. 024001-21